The Carnegie RR Lyrae Program: mid-infrared period-luminosity relations of RR Lyrae stars in Reticulum

2021-02-11T12:25:35Z

Acceptance in OA@INAF

The Carnegie RR Lyrae Program: mid-infrared period-luminosity relations of RR

Lyrae stars in Reticulum

Title

Muraveva, Tatiana; GAROFALO, Alessia; Scowcroft, Victoria; CLEMENTINI,

Gisella; Freedman, Wendy L.; et al.

Authors

10.1093/mnras/sty1959

DOI

http://hdl.handle.net/20.500.12386/30336

Handle

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY

Journal

480

Number

Advance Access publication 2018 July 23

The Carnegie RR Lyrae Program: mid-infrared period–luminosity

relations of RR Lyrae stars in Reticulum

Tatiana Muraveva,

Alessia Garofalo,

Victoria Scowcroft,

Gisella Clementini,

Wendy L. Freedman,

Barry F. Madore

and Andrew J. Monson

1_{INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti, 93/3, Bologna 40129, Italy} 2_{Universit´a di Bologna, Dipartimento di Fisica e Astronomia, Via Piero Gobetti 93/2, I-40129, Bologna, Italy} 3_{Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK}

4_{Observatories of the Carnegie Institution of Washington, 813 Santa Barbara St., Pasadena, CA 91101, USA} 5_{Department of Astronomy and Astrophysics, University of Chicago, 5640 S Ellis Ave, Chicago, IL 60637, USA}

6_{Department of Astronomy and Astrophysics, The Pennsylvania State University, 403 Davey Lab, University Park, PA 16802, USA}

Accepted 2018 July 19. Received 2018 July 13; in original form 2018 April 23

A B S T R A C T

We analysed 30 RR Lyrae stars (RRLs) located in the Large Magellanic Cloud (LMC) globular cluster Reticulum that were observed in the 3.6 and 4.5 μm passbands with the Infrared Array Camera (IRAC) on board the Spitzer Space Telescope. We derived new mid-infrared (MIR) period–luminosity (PL) relations. The zero points of the PL relations were estimated using the trigonometric parallaxes of five bright Milky Way RRLs measured with the Hubble Space

Telescope (HST) and, as an alternative, we used the trigonometric parallaxes published in the

first Gaia data release (DR1), which were obtained as part of the Tycho-Gaia Astrometric Solution (TGAS) and the parallaxes of the same stars released with the second Gaia data release (DR2). We determined the distance to Reticulum, using our new MIR PL relations, and found that distances calibrated on the TGAS and DR2 parallaxes are in a good agreement and, generally, smaller than distances based on the HST parallaxes, although they are still consistent within the respective errors. We conclude that Reticulum is located∼3 kpc closer to us than the barycentre of the LMC.

Key words: stars: distances – stars: variables: RR Lyrae – globular clusters: individual: Retic-ulum – Magellanic Clouds.

1 I N T R O D U C T I O N

Knowledge of the distances to celestial objects is crucially impor-tant to all branches of astronomy. In order to estimate distances, as-tronomers have developed a number of different techniques. Some methods such as the trigonometric parallax are based on geometric principles, while others invoke the use of primary distance indicators such as Cepheids or RR Lyrae (RRL) variables, which in turn are calibrated using the geometric methods. RRL stars (RRLs) are radi-ally pulsating variables that are abundant in globular clusters (GCs) and in the halos of galaxies. The RRLs are old (>10 Gyr), low-mass (∼0.6–0.8 M) core-helium-burning stars that lie within the instability strip crossing the horizontal branch (HB) of the colour-magnitude diagram (CMD). They are divided into those pulsating in the fundamental (RRab) and first-overtone (RRc) modes, and both modes simultaneously (RRd). RRLs are an excellent tool to estimate distances in the Milky Way (MW) and to Local Group

_{E-mail:tatiana.muraveva@inaf.it}

galaxies because they conform to an optical luminosity-metallicity (MV− [Fe/H]) relation and to infrared period–luminosity (PL) and

PL-metallicity (PLZ) relations. The near-infrared (NIR) and mid-infrared (MIR) PL relations of RRLs have several advantages in comparison with the visual MV− [Fe/H] relation; specifically, a

milder dependence of the luminosity on metallicity and lower sen-sitivity to interstellar extinction. Indeed, extinction in the 3.6 and 4.5 μm bands is much reduced with respect to the visual passband (A3.6∼ 0.06AVand A4.5∼ 0.05AV; Monson et al.2012).

Further-more, the intrinsic scatter of the RRL (and Cepheid) PL relations de-creases with increasing the wavelength (Madore & Freedman2012) and the MIR light curves of RRLs (and Cepheids) have smaller am-plitudes than in the optical passbands, hence the determination of the mean magnitudes is easier and more precise. Moreover, a num-ber of studies also indicate that the MV− [Fe/H] relation may be

not linear (Caputo et al.2000; Rey et al.2000; Bono et al.2003), which is not the case for the MIR PL relations.

In order to study the RRL PL relation in the MIR passbands, it is necessary to have a statistically significant sample of variables with firmly established classification, well-known periods and

ac-C

curately determined mean MIR magnitudes. In particular, for this study of the MIR RRL PL relation we selected Reticulum, an old GC located in the Large Magellanic Cloud (LMC),∼ 11 deg away from the centre of its host galaxy (Demers & Kunkel1976). The RRL population of Reticulum has been studied by several authors. Demers & Kunkel (1976) discovered 22 RRLs in the cluster. Walker (1992) later identified 10 additional RRLs bringing the total num-ber of RRLs in Reticulum to 32 (22 RRab, 9 RRc, and 1 candidate RRd star). Ripepi et al. (2004) combined Walker (1992)’s photom-etry with their observations in the B and V passbands and detected double-mode behaviour in four of the previously discovered RRL variables.

NIR K-band light curves with approximately 12 phase points per source were published by Dall’Ora et al. (2004) for 30 of the RRLs in Reticulum. Mean K-band magnitudes were estimated by fitting the light curves with templates from Jones, Carney & Fulbright (1996) and then used to build a NIR K-band PL relation.

Most recently, in their optical study of Reticulum, Kuehn et al. (2013) identified a secondary pulsation mode in two RRLs changing their classifications from RRc to RRd. Hence, the RRLs in Reticu-lum now list 22 RRab, 4 RRc, and 6 RRd stars. Kuehn et al. (2013) estimated an accuracy of 10−5 days for the periods of the RRab and RRc variables and an uncertainty approximately one order of magnitude larger for the periods of the RRd stars.

All RRLs in the cluster share the same reddening, distance, and metallicity (Kuehn et al.2013). Additionally, the majority of the RRLs in the cluster are located in uncrowded fields, which allows us to accurately determine their mean magnitudes in the 3.6 and 4.5 μm passbands.

In this paper we present the first MIR PL relations defined, using RRLs in a system outside our Galaxy, and compare them with empirical studies of the MIR PL relation based on field and GC MW RRLs (Madore et al.2013; Dambis, Rastorguev & Zabolotskikh

2014; Klein et al.2014; Neeley et al.2015; Gaia Collaboration

2017).

This study is part of the Carnegie RR Lyrae Program (CRRP, PI Freedman). CRRP is a Warm Spitzer Cycle 9 Exploration Science program, which uses the [3.6] and [4.5] channels of the Infrared Array Camera (IRAC; Fazio et al.2004) to study MW RRLs. In addition to the Reticulum observations described here, CRRP is also studying∼30 GCs in the MW and LMC, a sample of field RRLs in the MW (Monson et al.2017), and a large number of bulge RRLs. CRRP is complementary program to the Spitzer Merger History and Shape of the Halo Program (SMHASH; PI Johnston), which uses halo, stream, GC, and dwarf spheroidal galaxy (dSph) RRLs to examine the MW’s structure (e.g. Neeley et al.2017, Garofalo et al.2018, Hendel et al.2018).

To calibrate the zero point of our MIR PL relations, we used trigonometric parallaxes of five Galactic RRLs measured with the Fine Guide Sensor (FGS) on board the Hubble Space Telescope (HST) by Benedict et al. (2011). As a comparison, we have also derived the zero points for our MIR PL relations using the trigono-metric parallaxes for these same stars from the first data release (DR1) of the European Space Agency (ESA) mission Gaia (Gaia Collaboration2016a,2016b), which were computed as part of the Tycho-Gaia Astrometric Solution (TGAS; Lindegren et al.2016), in addition to the most recent Gaia parallaxes released with the sec-ond Data Release (DR2; Gaia Collaboration2018; Lindegren et al.

2018).

The paper is organized as follows. In Section 2.1, we describe the observations and data reduction, and provide MIR mean magnitudes and amplitudes of the 30 RRLs in Reticulum that form the basis

of this study. In Section 3 we present the results of fitting the RRL PL relations in the 3.6 and 4.5 μm passbands and describe the calibration of their zero points. We also present a comparison of our PL relations with those in the literature. In Section 4 we use our new relations to estimate the distance to Reticulum. A summary of our results is provided in Section 5.

2 DATA

2.1 Observations and data reduction

The Spitzer Space Telescope (Werner et al. 2004) was launched on 2003 August 25 with the main goal of providing a unique, infrared view of the universe. It operates at wavelengths that have low sensitivity to extinction and it has proven to be a perfect tool for undertaking variable star studies (Freedman et al. 2011) as it was able to be scheduled so as to uniformly sample the light curves of variable stars providing high-precision stellar photometry that is very stable in its calibrated zero point.

Reticulum was observed during Cycle 9 of the Warm Spitzer mis-sion. The observations were executed on 2012 November 27 and consist of 12 epochs (each with individual exposures of∼31 s), simultaneously in the 3.6 and 4.5 μm passbands. The observations were distributed over a time interval of approximately 14 h, corre-sponding to the longest period RRL in the field.

This spacing and time coverage were chosen to optimize the phase coverage of the RRLs that were known to exist in the field. The total area mapped by the Spitzer observations of Reticulum is shown in Fig.1. The cluster is located at the intersection of the Spitzer [3.6] and [4.5] fields of view. The elongated field shape is owing to the dither pattern of the observations. While ‘on target’ observations are taken in one IRAC band, adjacent ‘off target’ observations are taken simultaneously in the other passband. Colour information is available for the central region, and single band data are available for the two side regions. We outlined the cluster with a black circle of radius equal to two half-light radii of Reticulum (2.35 arcmin; Bica et al.1999).

For each epoch a single-epoch mosaic image was created from the individual basic calibrated data (BCD) frames provided by the Spitzer Science Center, using MOPEX (Makovoz & Khan

2005). Mosaicked location-correction images for each epoch were also created. Point-spread function (PSF) photometry of the mo-saicked images was performed using the daophot-allstar-allframepackages (Stetson1987,1994). First, each image mo-saic was converted from MJy/sr to counts (DN) using the con-version factor (MJy/sr per DN/sec) and the exposure time pro-vided in the image header. On each epoch the PSF model was created selecting∼50 stars uniformly distributed across the frames (with 30 per cent of stars in common among all epochs). Retic-ulum has an uncrowded nature, our PSF stars are bright (m[3.6]

< 16.741 mag and m[4.5]<16.519 mag) and well isolated. With

the ALLSTAR algorithm we fitted the PSF model given by the PSF stars to the sources in each frame. The ALLFRAME routine was then used to perform the PSF photometry simultaneously on images at all epochs. The ALLFRAME catalogue provides instru-mental magnitudes for each star in the images with an arbitrary zero-point value.

To transform the photometry to standard IRAC Vega magnitudes, we selected 53 stars in the 3.6 μm and 55 stars in the 4.5 μm images, homogeneously distributed over the frames. We obtained aperture photometry of these stars, calibrated to the zero-point magnitude

Figure 1. Coverage of the 3.6 μm (magenta box) and 4.5 μm (cyan box) images of Reticulum. The cluster is located at the intersection of the two fields of view (black dashed line). The black circle shows the location of the cluster, and has a radius equal to two half-light radii of Reticulum (2.35 arcmin; Bica et al. 1999).

(zmag) provided in the IRAC handbook,1 _{which was then}

com-pared with theDAOPHOTinstrumental magnitudes in order to derive

the final calibrated magnitudes. Point source photometry requires additional array location-dependent photometric corrections. These corrections improve the photometric accuracy for IRAC data by taking into account scattering and distortions, but above all, varia-tions in the effective central wavelengths of the IRAC filters, which depend on the angle of incidence. The final photometric catalogues contain 4068 and 3379 sources in 3.6 and 4.5 μm, respectively. The 3.6 μm photometry spans the range from 21.57 to 9.23 mag, while the 4.5 μm photometry ranges from 21.11 to 10.36 mag, where upper limits are owing to saturation. Uncertainties in the 3.6 μm magnitudes range from 0.003 to 0.100 mag, while uncertainties in the 4.5 μm magnitudes span the range from 0.005 to 0.100 mag. The typical uncertainties at the level of the Reticulum HB (∼18 mag) are 0.017 and 0.022 mag in the 3.6 μm and 4.5 μm bands, respectively. Typical uncertainties of the photometry per magnitude bin are shown in Fig.2. We remind the reader that the uncertainties in magnitudes calculated as part of the data reduction procedure usingALLFRAME software can be significantly underestimated.

We used DAOMASTER and DAOMATCH to combine the 3.6 μm and 4.5 μm catalogues and obtained a list of 1284 sources for which photometry is available in both passbands. The [3.6] versus ([3.6]–[4.5]) CMD of these sources is shown in Fig.2, where red dots represent the RRLs. The MIR CMD has its limitations owing to low sensitivity of the luminosity in the MIR passbands to the temperature. Although the HB is not very well distinguished from the Red Giant Branch (RGB) in these passbands, the RRLs in Fig.2

tracing a well-defined population, in both colour and magnitude allow us to clearly identify the location of the cluster HB.

Jeon et al. (2014) provide B, V photometry for 766 sources in Reticulum. We crossmatched the catalogue of Jeon et al. (2014)

1_{http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandb}

ook, Chapter 4

with our MIR catalogue of 1284 sources and found 364 objects in common. Information on these sources with the apparent B and V magnitudes provided by Jeon et al. (2014) and MIR magnitudes obtained here is presented in Table1.

The [3.6], (V− [3.6]) and [3.6], (B − [3.6]) CMDs of these 364 sources are shown in the left-hand panels of Fig. 3. The cluster RGB and the HB are well defined, demonstrating the quality of our photometry and the accuracy of the calibration procedure.

2.2 RRLs in Reticulum

In this paper we study the MIR (3.6 and 4.5 μm) time-series pho-tometry of the Reticulum RRLs. We cross-matched our catalogue of 1284 sources with 3.6 and 4.5 μm photometry (see Section 2.1) against 32 RRLs from Kuehn et al. (2013) catalogue. Thirty RRLs from the Kuehn et al. sample were identified, while we were not able to find the counterparts of V20 and V31 in our catalogue since these objects belong to non-resolved elongated groups of sources in our data. The 30 RRLs (20 RRab, 4 RRc, and 6 RRd stars) are listed in Table2adopting the identification number, coordinates, period, and classification in type of Kuehn et al. (2013). Periods range from 0.296 27 to 0.656 96 d. For the 6 RRd stars we list only the first-overtone periodicity. The positions of the 30 RRLs in the Spitzer images is shown in Fig.4with the RRab, RRc, and RRd stars marked by red, blue, and black open circles, respectively. The 3.6 and 4.5 μm light curves are presented in Fig.5.

The epoch of maximum light (Column 7 in Table2) is the same for the 3.6 and 4.5 μm passbands for all sources except for V10. The 4.5 μm data points of this star had to be shifted by∼0.4 in phase when folded according to the epoch in the 3.6 μm pass-band. For a large majority of the RRLs, we determined mean 3.6 and 4.5 μm magnitudes by Fourier fitting the light curves with the GRaphical Analyzer of TImes Series package (GRATIS, custom

soft-ware developed at the Observatory of Bologna by P. Montegriffo; see e.g. Clementini et al.2000). The blue lines in Fig.5show the best-fitting models obtained withGRATIS, while dots represent the

Figure 2. CMD ([3.6] versus [3.6]−[4.5]) of 1284 sources in the field of Reticulum observed with Spitzer in both 3.6 and 4.5 μm passbands. RRLs are marked in red. Typical uncertainties of the photometry per magnitude

bin estimated withALLFRAMEare shown on the right.

data points retained in the analysis. We discarded obvious outliers, however after the cleaning procedure each source still has at least 11 data points, since model fitting with theGRATISsoftware needs a

minimum number of 11 data points to converge. Mean magnitudes listed in Table2are intensity averages. The standard deviation of the phase points from the fit was calculated considering only data points retained after removing outliers. The final uncertainty in the mean magnitude was calculated as the sum in quadrature of the fit uncertainty provided byGRATISand the photometric uncer-tainty (σ2

i/N), where σiis the photometric uncertainty of the

individual observations and N is the number of observations. Peak-to-peak amplitudes were also measured from the modelled light curves.

For very noisy light curves, such as those of V04 and V12 in both the 3.6 and 4.5 μm passbands, and V08, V24, V28, V32 in the 4.5 μm passband, GRATISfailed to fit a reliable model to the data. For these stars the mean magnitudes were calculated as the weighted mean of the intensity values of each data point then transformed back to magnitudes, while the uncertainty of the mean values was estimated as the standard deviation of the weighted mean. No amplitudes were calculated for these stars.

Columns from 8 to 11 in Table2provide mean magnitudes with related uncertainties and amplitudes in the 3.6 and 4.5 μm bands for the RRLs in our sample, when available. In Fig.6the amplitudes in the 3.6 and 4.5 μm bands are plotted versus period (Bailey dia-grams) for the samples of 28 and 24 RRLs, respectively, for which a determination of the amplitude from the light curve with GRATIS was possible. RRab, RRc, and RRd stars are shown by red circles, blue triangles, and green squares, respectively. As expected, the

RRab stars are well separated from RRc and RRd stars. The latter are located on the longest tail of the RRc period distribution. Con-sistently with the behaviour in the optical bands, the amplitude of the RRab stars decreases with increasing the period. Black crosses in Fig.6indicate three variables, V06, V14, and V23 that, according to Kuehn et al. (2013) possibly exhibit the Blazhko effect (Blako

1907), a modulation of both amplitude and shape of the light curve that typically occurs on time spans ranging from a few days to a few hundreds of days. These stars are marked with a ‘BL’ flag in Table2. The Blazhko effect should not affect significantly the mean magnitude we measured for the Blazhko RRLs because the time interval covered by our Spitzer observations (0.6 d) is short in com-parison with the typical Blazhko periods. Therefore, we retained the Blazhko variables in our analysis.

In order to clean our sample from blended sources and other contaminants, we analysed the position of the RRLs on the CMDs shown in Fig.3and the light curves (Fig.5) in combination with a visual inspection of the Spitzer images (Fig4). Fig.3demonstrates that most of the variables conform to the RRc stars (blue triangles) being bluer than the RRab variables (red filled circles), and the RRd stars (green squares) being located between RRc and RRab stars. However, for five objects, namely, V01, V08, V19, V23, and V24 the position on the CMD is unusual. V08 is far too bright and red in colour than the bulk of the RRab stars. The star light curves are noisy, especially in the 4.5 μm band (Fig.5). Inspection of the Spitzer images showed that V08 is among an unresolved group of sources. V01 is slightly separated and fainter than other RRLs in the CMD. Moreover, V01 has a significantly redder colour than it would be expected for its period (P= 0.51 d). The source is located on the upper edge of the 3.6 and 4.5 μm images and does not have a characteristic stellar shape on the images; hence, the PSF photometry is likely not reliable. Star V23 is potentially a Blazhko variable and is slightly fainter than other RRLs, but the 3.6 and 4.5 μm images and the star light curves do not have particular issues. The faint magnitude of this star may be consistent with its period (0.468 63 d) that is the shortest among all the RRab stars in our sample. It is worth noting that V23 has also the faintest V (Kuehn et al.2013) magnitudes among all RRLs in Reticulum, which, in combination with the results of this study, may prove that V23 is indeed the intrinsically faintest star in the sample. While the faint MIR magnitudes can be explained by the shortest period of this star among all the RRab stars in the sample, the reason of the faint visual magnitude is unclear. A high metallicity of V23 could potentially account for its faint V magnitude; however, it would contradict the general picture of Reticulum being monometallic cluster. Spectroscopical analysis of RRLs in Reticulum can shed light on this issue in the future. An alternative possibility is that V23 is a foreground RRL belonging to the field of the LMC. Star V19 has a too blue colour for an RRab variable. Moreover, it has a rather noisy light curve especially in the 4.5 μm band. Visual inspection of the images showed that V19 has a companion with 3.6 μm apparent magnitude 18 mag at a distance of 0.82 arcsec. The source is also located close to the edge of the 4.5 μm image. Both these issues may have affected the photometry. V24 appears too red in colour for an RRd star and has an extremely noisy light curve. The star is located at a distance of 3.79 arcsec from a 15 mag bright source, with which it is blended in both passbands. Finally, V28 and V32 are not separated from the other RRLs on the CMD, but have noisy light curves. V28 has a close companion (at a distance of 1.63 arcsec) of magnitude∼19 mag, while V32 is blended by a companion with magnitude 18 mag located at a distance of 1.87 arcsec. Two further stars (V04 and V12) have noisy light curves

Table 1. Main properties of the 364 sources in common between our MIR catalogues and the visual catalogues of Jeon et al. (2014).

ID Jeona _{ID Kuehn}b _{RA Dec. Flag}c _{B V [3.6]}d _[4.5]d

(J2000) (J2000) (mag) (mag) (mag) (mag)

Reticulum 225 V01 04:35:51.5 −58:51:03.4 1 19.771± 0.115 19.364± 0.084 17.751 17.826 Reticulum 279 V02 04:35:56.6 −58:52:32.1 1 19.467± 0.256 19.074± 0.205 17.717 17.678 Reticulum 269 V03 04:35:58.6 −58:53:06.7 1 19.385± 0.216 19.112± 0.163 17.997 17.932 Reticulum 272 V04 04:36:00.3 −58:52:50.0 1 19.307± 0.218 19.040± 0.178 17.963 17.974 Reticulum 250 V05 04:36:04.1 −58:52:29.3 1 19.440± 0.286 19.070± 0.239 17.669 17.611 Reticulum 294 V06 04:36:05.4 −58:51:47.8 1 19.495± 0.257 19.112± 0.216 17.747 17.826 Reticulum 220 V07 04:36:05.5 −58:50:51.8 1 19.185± 0.504 18.943± 0.421 17.821 17.978 Reticulum 288 V08 04:36:05.7 −58:49:34.3 1 19.421± 0.189 19.016± 0.129 16.841 16.602 Reticulum 237 V09 04:36:05.9 −58:50:24.5 1 19.306± 0.341 18.998± 0.286 17.726 17.756 Reticulum 273 V10 04:36:06.4 −58:52:12.6 1 19.365± 0.224 19.102± 0.200 17.960 18.002 Reticulum 259 V11 04:36:06.4 −58:51:48.7 1 19.280± 0.268 18.992± 0.228 17.913 18.037 Reticulum 243 V12 04:36:07.1 −58:50:54.9 1 19.175± 0.105 18.984± 0.084 18.124 18.157 Reticulum 255 V13 04:36:07.8 −58:51:46.8 1 19.531± 0.322 19.116± 0.255 17.773 17.743 Reticulum 244 V14 04:36:07.8 −58:51:44.4 1 19.427± 0.324 19.065± 0.246 17.683 17.792 Reticulum 262 V15 04:36:09.1 −58:52:25.8 1 19.453± 0.162 19.152± 0.142 17.980 18.050 Reticulum 233 V16 04:36:09.8 −58:52:50.7 1 19.254± 0.436 19.002± 0.340 17.814 17.823 Reticulum 298 V17 04:36:10.3 −58:53:13.6 1 19.267± 0.419 18.937± 0.313 17.754 17.850 Reticulum 261 V18 04:36:10.6 −58:49:50.2 1 19.511± 0.308 19.145± 0.263 17.733 17.760 Reticulum 227 V19 04:36:11.9 −58:49:18.1 1 19.070± 0.449 18.754± 0.332 18.029 17.952 Reticulum 287 V21 04:36:12.3 −58:49:24.3 1 19.578± 0.283 19.155± 0.217 17.798 17.663 Reticulum 276 V22 04:36:13.4 −58:52:32.0 1 19.309± 0.363 18.967± 0.260 17.801 17.989 Reticulum 258 V23 04:36:13.8 −58:51:19.0 1 19.790± 0.134 19.392± 0.120 18.034 18.170 Reticulum 248 V24 04:36:17.3 −58:51:26.6 1 19.377± 0.273 19.106± 0.229 17.660 17.803 Reticulum 277 V25 04:36:17.4 −58:53:02.6 1 19.367± 0.255 19.134± 0.178 18.058 18.097 Reticulum 296 V26 04:36:18.5 −58:51:51.8 1 19.490± 0.155 19.065± 0.113 17.668 17.698 Reticulum 295 V27 04:36:18.7 −58:51:44.3 1 19.669± 0.263 19.243± 0.200 17.779 17.837 Reticulum 246 V28 04:36:19.2 −58:50:15.5 1 19.294± 0.223 19.043± 0.166 18.003 18.071 Reticulum 223 V29 04:36:20.1 −58:52:33.5 1 19.446± 0.381 19.058± 0.285 17.832 17.847 Reticulum 236 V30 04:36:20.2 −58:52:47.7 1 19.582± 0.268 19.119± 0.284 17.837 17.835 Reticulum 268 V32 04:36:32.0 −58:49:53.2 1 19.319± 0.207 19.016± 0.160 17.953 18.003 Reticulum 5 – 04:35:52.8 −58:49:40.5 0 13.825± 0.017 13.191± 0.020 12.034 11.625 Reticulum 12 – 04:35:50.9 −58:52:55.7 0 15.661± 0.014 14.151± 0.014 10.144 10.805 Reticulum 19 – 04:36:19.4 −58:54:20.6 0 15.950± 0.006 15.081± 0.016 13.103 13.121 Reticulum 23 – 04:36:12.3 −58:54:04.4 0 16.149± 0.009 15.294± 0.017 12.992 12.977 Reticulum 26 – 04:36:22.5 −58:49:08.2 0 16.212± 0.008 15.541± 0.016 13.787 13.880 Notes:

a_{Identification from Jeon et al. (2014).}

b_{Identification from Kuehn et al. (2013).}

c_{A ‘1’ flag indicates objects from Jeon et al. (2014) that are classified as RRLs by Kuehn et al. (2013), while a ‘0’ flag indicates stars that do not have a}

counterpart in the Kuehn et al. (2013) catalogue of RRLs.

d_{Mean magnitudes in the Spitzer passbands obtained as a result of the data-reduction procedure; uncertainties in magnitudes are not provided, since allframe}

is known to provide underestimated uncertainties.

This table is published in its entirety at the CDS; a portion is shown here for guidance with respect to its form and content. but their images and the position on the CMD do not indicate any

clear issue. V04 is a double mode pulsator and V12 is the shortest period (and hence faint) RRL in our sample. This may explain the noisy light curves. To summarize, the following six stars were dropped: V01, V08, V19, V24, V28, and V32, while V04 and V12 were retained to avoid biasing our derivation of the PL relations by brighter/longer period stars. Therefore in the following analysis we used a subsample of 24 RRLs consisting of 17 RRab, 3 RRc, and 4 RRd stars.

3 M I R PL R E L AT I O N S 3.1 Slope of the PL relations

A number of studies indicate a decreasing dependence of the RRL luminosity on metallicity when moving to longer wavelengths. A rather small metallicity dependence of the K-band luminosity has

been found by empirical studies (see table 3 in Muraveva et al.2015

for a summary), while theoretical studies suggest the dependence of the K-band luminosity on metallicity to be non-negligible (Bono et al.2001; Catelan, Pritzl & Smith2004; Marconi et al.2015), which recently was also confirmed by Braga et al. (2018) based on the analysis of RRLs in the GC ω Cen. The MIR PL relations were derived neglecting the dependence of the luminosity on metallicity in a number of studies (e.g. Madore et al.2013; Klein et al.2014; Neeley et al. 2015, Gaia Collaboration 2017). At the same time, some theoretical (Neeley et al.2017) and empirical (Dambis et al.

2014; Sesar et al. 2017; Muraveva et al.2018) studies suggest a non-negligible dependence of the MIR luminosity on metallicity (metallicity slope≥0.1 mag/dex; see Table3). The RRLs analysed in this paper share the same metal abundance; hence, they cannot be used to estimate the effect of metallicity on the MIR PL relations; however, we urge the reader to keep in mind a possible dependence of the MIR luminosity on metallicity.

Figure 3. Left-hand panels: CMDs ([3.6] versus V− [3.6] and [3.6] versus B − [3.6], in the upper- and bottom-left panels, respectively) of 364 sources in

the field of the Reticulum cluster that we observed with Spitzer. B and V photometry for these objects is available from Jeon et al. (2014). Red circles, blue

triangles, and green squares represent RRab, RRc, and RRd stars, respectively. Right-hand panels: Zoom-in of the region populated by RRLs in the CMDs shown in the left-hand panels. For clarity reasons, we do not plot the RRL V08 in the right-hand panels of the figure.

Before we fit the MIR PL relations, we corrected the apparent 3.6 and 4.5 μm magnitude of the RRLs in our sample for interstel-lar extinction. Reddening E(B− V) values towards the Reticulum cluster reported in the literature are generally low but controversial, and span the range from 0.016 (Schlegel, Finkbeiner & Davis1998) to 0.06 mag (Marconi et al.2002). The range of reddening values for Reticulum reported in the literature implies an uncertainty in the V magnitude, and therefore its distance estimation, of∼0.14 mag. The situation improves significantly when moving to the MIR passbands, as the full range in reddening values reported in the lit-erature causes uncertainties of only 0.009 and 0.007 mag in the 3.6 and 4.5 μm bands, respectively, which are negligible in comparison

with other uncertainties. In this paper, we adopt the reddening value E(B− V) = 0.03 ± 0.02 mag from Walker (1992), who estimated the reddening from the colours of the RRab stars at minimum light and Sturch’s method (Sturch1966; Walker1990). This value of the reddening is close to the mean of the reddening estimates in the literature. The following relations obtained by Monson et al. (2012) based on the reddening laws from Cardelli, Clayton & Mathis (1989) and Indebetouw et al. (2005) were used to estimate the interstellar extinctions:

Ta b le 2 . Identification and properties of the 3 0 RRLs in R eticulum for which we analysed the Spitzer 3.6 and 4.5 μ m time-series data. ID RA Dec. T ype P Log( P ) E poch a [3.6] b Amp [3.6] [4.5] b Amp [4.5]  Flag c (J2000) (J2000) (days) (HJD) (mag) (mag) (mag) (mag) (mas) V01 04:35:51.5 − 58:51:03.4 RRab 0 .50993 − 0.29249 56258.032 17.750 ± 0.109 0.410 17.825 ± 0.185 0.217 − 0.18 ± 0.19 1 V02 04:35:56.6 − 58:52:32.1 RRab 0 .61869 − 0.20853 56257.239 17.719 ± 0.047 0.153 17.603 ± 0.108 0.238 0.13 ± 0.18 0 V03 04:35:58.6 − 58:53:06.7 RRd 0.35350 − 0.45161 56257.592 18.015 ± 0.068 0.204 17.918 ± 0.077 0 .244 − 0.02 ± 0.18 0 V04 04:36:00.3 − 58:52:50.0 RRd 0.35320 − 0.45198 56268.034 17.958 ± 0.056 – 1 7.908 ± 0.103 – 0 .20 ± 0.18 0 V05 04:36:04.1 − 58:52:29.3 RRab 0 .57185 − 0.24272 56257.444 17.711 ± 0.042 0.249 17.617 ± 0.063 0.322 0.14 ± 0.21 0 V06-BL 04:36:05.4 − 58:51:47.8 RRab 0 .59526 − 0.22529 56257.439 17.775 ± 0.041 0.276 17.862 ± 0.094 0.363 − 0.01 ± 0.18 0 V07 04:36:05.5 − 58:50:51.8 RRab 0 .51044 − 0.29206 56257.550 17.849 ± 0.052 0.280 18.010 ± 0.097 0.255 0.43 ± 0.23 0 V08 04:36:05.7 − 58:49:34.3 RRab 0 .64496 − 0.19047 56257.608 16.796 ± 0.055 0.144 16.558 ± 0.097 – 0 .19 ± 0.20 1 V09 04:36:05.9 − 58:50:24.5 RRab 0 .54496 − 0.26364 56257.824 17.739 ± 0.047 0.166 17.732 ± 0.069 0.261 − 0.09 ± 0.17 0 V10 04:36:06.4 − 58:52:12.6 RRc 0.35256 − 0.45277 56257.813 17.967 ± 0.044 0.185 17.964 ± 0.077 0 .166 0.02 ± 0.18 0 V11 04:36:06.4 − 58:51:48.7 RRd 0.35540 − 0.44928 56257.987 17.904 ± 0.045 0.160 18.000 ± 0.103 0 .160 0.02 ± 0.21 0 V12 04:36:07.1 − 58:50:54.9 RRc 0.29627 − 0.52831 56258.029 18.108 ± 0.056 – 1 8.184 ± 0.104 – 0 .11 ± 0.19 0 V13 04:36:07.8 − 58:51:46.8 RRab 0 .60958 − 0.21497 56257.456 17.782 ± 0.039 0.175 17.716 ± 0.055 0.209 − 0.05 ± 0.18 0 V14-BL 04:36:07.8 − 58:51:44.4 RRab 0 .58661 − 0.23165 56257.679 17.681 ± 0.044 0.288 17.728 ± 0.052 0.289 0.10 ± 0.19 0 V15 04:36:09.1 − 58:52:25.8 RRd 0.35430 − 0.45063 56257.790 17.972 ± 0.053 0.138 18.040 ± 0.115 0 .200 − 0.04 ± 0.20 0 V16 04:36:09.8 − 58:52:50.7 RRab 0 .52290 − 0.28158 56257.913 17.829 ± 0.056 0.246 17.827 ± 0.073 0.210 0.10 ± 0.19 0 V17 04:36:10.3 − 58:53:13.6 RRab 0 .51241 − 0.29038 56257.599 17.767 ± 0.045 0.302 17.757 ± 0.076 0.340 0.39 ± 0.20 0 V18 04:36:10.6 − 58:49:50.2 RRab 0 .56005 − 0.25177 56257.751 17.701 ± 0.050 0.291 17.722 ± 0.066 0.247 − 0.50 ± 0.19 0 V19 04:36:11.9 − 58:49:18.1 RRab 0 .48485 − 0.31439 56257.635 18.046 ± 0.074 0.360 17.981 ± 0.155 0.603 0.30 ± 0.20 1 V21 04:36:12.3 − 58:49:24.3 RRab 0 .60700 − 0.21681 56257.933 17.760 ± 0.065 0.292 17.665 ± 0.089 0.263 − 0.05 ± 0.20 0 V22 04:36:13.4 − 58:52:32.0 RRab 0 .51359 − 0.28938 56257.851 17.844 ± 0.072 0.291 17.880 ± 0.077 0.351 − 0.35 ± 0.22 0 V23-BL 04:36:13.8 − 58:51:19.0 RRab 0 .46863 − 0.32917 56258.153 18.082 ± 0.070 0.244 18.245 ± 0.114 0.366 − 0.01 ± 0.21 0/1 V24 04:36:17.3 − 58:51:26.6 RRd 0.34750 − 0.45905 56257.864 17.661 ± 0.067 0.089 17.780 ± 0.119 – − 0.12 ± 0.19 1 V25 04:36:17.4 − 58:53:02.6 RRc 0.32991 − 0.48160 56258.020 18.080 ± 0.065 0.191 18.073 ± 0.077 0 .242 0.11 ± 0.19 0 V26 04:36:18.5 − 58:51:51.8 RRab 0 .65696 − 0.18246 56257.940 17.653 ± 0.042 0.100 17.720 ± 0.087 0.222 − 0.11 ± 0.19 0 V27 04:36:18.7 − 58:51:44.3 RRab 0 .51382 − 0.28919 56257.982 17.852 ± 0.052 0.350 17.900 ± 0.071 0.331 − 0.31 ± 0.22 0 V28 04:36:19.2 − 58:50:15.5 RRc 0.31994 − 0.49493 56257.701 17.952 ± 0.065 0.163 18.056 ± 0.141 – 0.23 ± 0.22 1 V29 04:36:20.1 − 58:52:33.5 RRab 0 .50815 − 0.29401 56257.682 17.872 ± 0.067 0.348 17.818 ± 0.078 0.440 − 0.14 ± 0.20 0 V30 04:36:20.2 − 58:52:47.7 RRab 0 .53501 − 0.27164 56257.725 17.769 ± 0.047 0.242 17.750 ± 0.076 0.301 − 0.19 ± 0.18 0 V32 04:36:32.0 − 58:49:53.2 RRd 0.35230 − 0.45309 56257.750 17.903 ± 0.060 0.060 18.083 ± 0.208 – 0.07 ± 0.22 1 Notes : aEpoch o f m aximum light (HJD-2400000) of the 3 .6 μ m light curv e. The epoch o f m aximum light is the same in the 3.6 μ m and 4.5 μ m p assbands for all sources ex cept for V10. The 4 .5 μ m d ata of this star had to be shifted by 0.4 in phase. bMagnitudes in the Spitzer passbands obtained by fitting a model to the data with the GRA T IS softw are. cA ‘0’ in column 13 flags objects that were used to fit the PL relations whereas a ‘1’ flags stars that w ere d iscarded. A ‘0/1’ flags V23 b ecause the star w as u sed to fi t the the 3 .6 μ m PL relation, b u t w as automatically rejected by the 3 σ clipping procedure w hen fi tting the 4.5 μ m PL relation. See the te xt for d etails.

Figure 4. Zoom-in of Fig.1showing the region of the Reticulum cluster. Open red, blue, and black circles indicate the Reticulum RRab, RRc, and RRd stars, respectively.

A[4.5]= 0.156 ∗ E(B − V ) (2)

and derive the extinction values of 0.006 and 0.005 mag in the 3.6 and 4.5 μm bands, respectively. The reddening-corrected 3.6 and 4.5 μm magnitudes were then used to fit the MIR PL relations.

We used the weighted least squares method and a 3σ clipping pro-cedure to fit the PL relations in the 3.6 and 4.5 μm passbands defined by the RRLs in our sample. Following Dall’Ora et al. (2004), the RRd variables were treated as RRc stars. Periods of first-overtone and double-mode RRLs were then ‘fundamentalized’ according to the relation:

log(P )= log(Pfo)+ 0.127, (3)

where Pfois the first-overtone period. The following PL relations

were derived:

[3.6]= (−2.15 ± 0.23) × log(P ) + (17.23 ± 0.06) (4) with rms=0.06 mag,

[4.5]= (−2.47 ± 0.32) × log(P ) + (17.14 ± 0.09) (5) with rms=0.07 mag, where [3.6] and [4.5] are the absorption-corrected apparent magnitudes. The best-fitting relations are shown in Fig.7, plotting with filled circles the RRLs used to fit the PL relations, highlighting in red the Blazhko variables and marking with green and blue triangles six stars that were discarded when fit-ting the 4.5 and 3.6 μm PL relations, respectively, according to the analysis described in Section 2.2. We also note that star V23 was automatically rejected by the 3σ clipping procedure when fitting the PL relation in the 4.5 μm passband, which therefore is based

on only 23 RRLs. We have marked this star with an empty circle in Fig.7.

3.2 Zero-point calibration

To transform to absolute magnitudes, the apparent magnitudes in equations (4) and (5), we must calibrate the zero points of the PL relations. Trigonometric parallaxes are the only direct method to estimate distances and infer absolute magnitudes. Until recently ac-curate parallax measurements were available for only a handful of RRLs. The ESA mission Hipparcos (van Leeuwen2007and refer-ences therein) measured the parallax for more than 100 RRLs in the solar neighbourhood, but with errors larger than 30 per cent, except for RR Lyr itself, the bright prototype of the RRL class, for which the error is∼18 per cent. Benedict et al. (2011) measured accurate trigonometric parallaxes of five Galactic RRLs (among which is RR Lyr) using the FGS on board the HST. However, we note that the Hipparcos and HST parallax of RR Lyr are only in marginal agree-ment. Therefore, a direct calibration of the RRL distance relations has so far been hampered not only by a lack of good-quality paral-lax data, but also by the few independent measurements presently available providing inconsistent results. Gaia, the ESA cornerstone mission launched on 2013 December 19 is measuring trigonometric parallaxes (along with positions, proper motions, photometry, and physical parameters) for over one billion stars in the MW and be-yond. Gaia will revolutionize the field by providing end-of-mission parallax measurements with about 10 μas uncertainty for RRLs brighter than V∼ 12–13 mag.

Figure 5. Spitzer MIR light curves of the RRLs in Reticulum analysed in this study. Black and red dots represent the 3.6 μm and 4.5 μm data points,

Figure 6. Period-amplitude (Bailey) diagrams in the 3.6 (upper panel) and 4.5 μm (lower panel) passbands defined, respectively, by 28 and 24 RRLs in our sample, for which the determination of the amplitude with GRATIS was possible. Red circles, blue triangles, and green squares represent RRab, RRc, and RRd stars, respectively. Stars possibly affected by the Blazhko effect are marked by black crosses.

A first anticipation of Gaia promise in this field came on 2016 September 14 with the publication of the Gaia DR1 (Gaia Collab-oration2016a,2016b). The DR1 catalogue contains parallaxes for about 2 million stars in common between Gaia and the Hipparcos and Tycho-2 catalogues, computed as part of the TGAS (Lindegren et al.2016). The TGAS sample includes parallaxes for 364 MW RRLs, of which a fraction was used by Gaia Collaboration (2017) to calibrate the RRL visual MV− [Fe/H] relation and the infrared

PL and PLZ relations. Even though TGAS parallaxes for RRLs show an extraordinary improvement with respect to Hipparcos (see e.g. fig. 23 in Gaia Collaboration2017), their uncertainties are still large, spanning from 0.209 to 0.967 mas in range. The situation improved significantly on 2018 April 25, when the Gaia

trigono-metric parallaxes for a much larger sample of RRLs using only Gaia measurements were published in Gaia DR2. The Gaia DR2 catalogue comprises positions and multiband photometry for∼1.7 billion sources alongside the parallaxes and proper motions for∼1.3 billion sources (Gaia Collaboration2018). Furthermore, Gaia DR2 published a catalogue of more than∼500 000 variables of differ-ent types (Holl et al.2018), including 140 784 RRLs (Clementini et al.2018). The dramatic improvement brought by Gaia DR2 for RRLs, compared to its precursors Hipparcos and Gaia DR1, is demonstrated in fig. 6 of Muraveva et al. (2018).

We retrieved Gaia DR2 parallaxes for the 30 RRLs in Reticulum from the Gaia Archive website.2_{The resulting values are shown}

in Table2. Unfortunately, owing to the large distances and faint magnitudes of the RRLs in Reticulum (mean Gaia G-band magni-tude <G >= 19.02 mag), 50 per cent of the Reticulum RRLs have negative parallax values, while the remaining 15 RRLs have large relative parallax uncertainties (<σ/ >= 2.7). Consequently,

the Reticulum sample is unsuitable for calibrating the zero points of the RRL MIR PL relations. Instead we adopt the slope of the MIR PL relations derived for the RRLs in Reticulum (equations 4 and 5), and calibrate the zero points using the parallaxes of RRLs in the MW. We selected the HST parallaxes for the five RRLs in Benedict et al. (2011) and the TGAS and Gaia DR2 parallaxes for the same stars, and then compared the resulting PL relations with the literature relations (Section 3.3).

Apparent magnitudes in the Spitzer passbands for the five MW RRLs with HST parallaxes published by Benedict et al. (2011), namely, RZ Cep, XZ Cyg, SU Dra, RR Lyr, and UV Oct, are provided in table 5 of Neeley et al. (2015), HST parallaxes are taken from table 8 of Benedict et al. (2011) and the TGAS and Gaia DR2 parallaxes are retrieved from Gaia Archive website but they are also listed in table 1 of Gaia Collaboration (2017), for TGAS, and table 2 of Muraveva et al. (2018), for the DR2 parallaxes. Periods of these five stars range from 0.3086 to 0.6604 days and metallicities span the range from [Fe/H]=−1.80 to −1.41 dex on the Zinn & West (1984) metallicity scale. The metallicity of the RRLs in Reticulum is [Fe/H]=−1.66 dex on the Zinn & West (1984) metallicity scale (Mackey & Gilmore2004), which is well within the range spanned by Benedict et al. (2011)’s variables. The logarithm of the RRc star RZ Cep was ‘fundamentalized’ (equation 3).

The DR2 parallax for RR Lyr itself has a large negative value (−2.61 ± 0.61 mas) and is clearly wrong (Arenou et al.2018; Gaia Collaboration2018). Hence, or this star we adopt the TGAS parallax instead of the DR2 measure.

As widely discussed in Gaia Collaboration (2017), the direct transformation of parallaxes into absolute magnitudes using the relation:

M= m0+ 5 log  − 10, (6)

where M is the star absolute magnitude, m0is the dereddened

ap-parent magnitude, and  is the parallax in mas and has significant disadvantages. While errors in parallaxes are approximately Gaus-sian and symmetrical, the logarithm in equation (6) causes the errors in absolute magnitudes to become asymmetrical leading to biased results (Gaia Collaboration2017; Luri et al.2018). Furthermore, using this method means that negative parallaxes cannot be trans-formed into absolute magnitudes, which causes a truncation bias in the data. In order to circumvent both issues, it is advisable to operate

Table 3. Comparison of MIR period–luminosity–(metallicity) relations in the form α× log (P) + β × [Fe/H] + ZP.

Reference Band α β ZP ZP ZP ZP from

from HST  from TGAS  from Gaia DR2  other sources

PL relations (W1, [3.6])

This paper 3.6 μm −2.150 ± 0.230 – −1.190 ± 0.050 −1.090 ± 0.090 −1.080 ± 0.03 –

Madore et al. (2013) W1 −2.440 ± 0.950 – −1.260 ± 0.250 – – −

Klein et al. (2014) RRab W1 −2.380 ± 0.200 − – – – −1.113 ± 0.013a

Klein et al. (2014) RRc W1 −1.640 ± 0.620 − – – – −1.042 ± 0.031a Neeley et al. (2015) 3.6 μm −2.332 ± 0.106 − −1.176 ± 0.080 – − −1.054 ± 0.020b Gaia Collaboration (2017) W1 −2.440 – – −1.210 ± 0.040 – – Neeley et al. (2017) 3.6 μm −2.300 ± 0.110 – −1.112 ± 0.089 – – – PLZ relations (W1, [3.6]) Dambis et al. (2014) W1 −2.381 ± 0.097 0.096± 0.021 −1.150 ± 0.077 – – −0.829 ± 0.093c Neeley et al. (2017) W1 −2.247 ± 0.018 0.180± 0.003 – − – −0.790 ± 0.007d Neeley et al. (2017) 3.6 μm −2.251 ± 0.018 0.180± 0.003 – – – −0.793 ± 0.007d Sesar et al. (2017) W1 −2.470+0.74_−0.73 0.150+0.09_−0.08 – −0.890+0.12_−0.10e _{– −} Muraveva et al. (2018) W1 −2.450+0.88_−0.82 0.160+0.10_−0.10 – – −0.910+0.36_−0.34 – PL relations (W2, [4.5]) This paper 4.5 μm −2.470 ± 0.320 – −1.290 ± 0.050 −1.200 ± 0.080 −1.200 ± 0.030 – Madore et al. (2013) W2 −2.550 ± 0.890 – −1.290 ± 0.230 – – –

Klein et al. (2014) RRab W2 −2.390 ± 0.200 – – – – −1.110 ± 0.013a

Klein et al. (2014) RRc W2 −1.700 ± 0.620 – – – – −1.057 ± 0.031a Neeley et al. (2015) 4.5 μm −2.336 ± 0.105 – −1.199 ± 0.080 – – −1.091 ± 0.020b Neeley et al. (2017) 4.5 μm −2.340 ± 0.100 – −1.139 ± 0.089 – – – PLZ relations (W2, [4.5]) Dambis et al. (2014) W2 −2.269 ± 0.127 0.108± 0.021 −1.105 ± 0.077 – – −0.776 ± 0.093c Neeley et al. (2017) W2 −2.237 ± 0.018 0.185± 0.003 – – – −0.784 ± 0.007d Neeley et al. (2017) 4.5 μm −2.239 ± 0.018 0.185± 0.003 – – – −0.785 ± 0.007d Sesar et al. (2017) W2 −2.400+0.84_−0.82 0.170+0.10_−0.09 – −0.947+0.11_−0.10e _{– −} Notes:

a_{Zero point derived from the M}

V− [Fe/H] relation of RRLs in combination with the HST parallaxes of RRLs in Benedict et al. (2011). b_{Zero point calibrated by adopting the distance modulus of the GC M4.}

c_{Zero point determined from the statistical-parallax analysis.} d_{Theoretically determined zero point.}

e_{Zero point is from table 1 and equation 4 in Sesar et al. (2017), assuming P}

ref= 0.52854 d and [Fe/H]ref= −1.4 dex.

directly in the parallax space, using, for instance, the reduced par-allax (Feast & Catchpole1997) or Astrometry-Based Luminosity (ABL; Arenou & Luri1999) method that is defined by the relation:

ABL= 100.2M _{= 10}0.2(α log P+ZP )_{=  10}0.2m0−2_{, (7)}

where M is the stars absolute magnitude, P is the period, m0 is

the dereddened apparent magnitude,  is the parallax in mas, α is the slope of the PL relation, for which we adopt the values we have derived from the RRLs in Reticulum (equations 4 and 5 in Section 3.1), and ZP is the PL zero point that we aim to determine. Using the ABLs instead of the absolute magnitudes for our RRL calibrators allows us to maintain symmetrical errors.3_{We also note}

that there is no need to apply any Lutz–Kelker correction (Lutz & Kelker1973) in our study, since individual measurements of objects that were chosen independently of their parallax values or relative errors in parallaxes are not affected by the Lutz Kelker bias (Arenou & Luri1999, Feast2002), and indeed the five RRLs in Benedict et al. (2011) were not selected based on their observed parallaxes.

A number of independent studies regarding a possible zero-point offset of Gaia DR2 parallaxes appeared recently in literature (e.g. Arenou et al.2018; Muraveva et al.2018; Riess et al.2018; Stassun & Torres2018; Zinn et al. 2018). All these studies indicate that

3_{More details on the use of the ABL method to fit the PL relation can be}

found in Gaia Collaboration (2017).

the Gaia DR2 parallaxes are systematically smaller with a mean estimated offset of = −0.03mas (Arenou et al.2018). Arenou et al. (2018) do not recommend to correct for this possible paral-lax offset individual star paralparal-laxes. Hence, we did not apply this correction to the five MW RRLs in our study.

The resulting 3.6 and 4.5 μm PL fits obtained by using the HST, TGAS, and DR2 parallaxes of the five MW RRL calibrators are shown in Figs8,9, and10, respectively. The fits were performed in parallax space, with parallaxes later transformed to absolute mag-nitudes in order to visualize the data on the PL plane. The slopes and zero points of the fits are listed in Table3.

3.3 Comparison with the literature

In recent years, several authors have studied the RRL MIR PL relations. These literature relations were calibrated either using the trigonometric parallaxes or by indirect methods to determine the zero point. Here, we briefly summarize these previous studies with a specific focus on the adopted calibration procedures. In Table3

we present the literature slopes and zero points alongside the values derived in this study.

Madore et al. (2013) used Wide-field Infrared Survey Explorer (WISE) observations in the [W1] (3.4 μm), [W2] (4.6 μm), and [W3] (12 μm) passbands for the four fundamental mode RRLs in Benedict et al. (2011) (they excluded the first-overtone star RZ

Figure 7. PL relations in the 3.6 and 4.5 μm passbands defined by the Reticulum RRLs. Filled circles represent RRLs used in the fit, with the Blazhko variables highlighted in red. Blue and green triangles show six RRLs that were discarded from the fit of, respectively, 3.6 and 4.5 μm PL relations as a result of the analysis of the light curves, position on the CMD, and Spitzer images (Section 2.2). Star V23 was also discarded from the fit of the 4.5 μm PL relation by 3σ clipping procedure and is marked with a red empty circle (see the text for details).

Figure 8. PL relations in the 3.6 (upper panel) and 4.5 μm (lower panel) passbands of five Galactic RRLs with the zero points calibrated using the HST parallaxes of Benedict et al. (2011). The logarithm of the period of RZ Cep was ‘fundamentalized’ (equation 3). Slopes are from this study (equations 4 and 5). The fits were performed in parallax space. See the text for details.

Figure 9. Same as in Fig.8, but with the PL zero points calibrated using TGAS parallaxes for the five Galactic RRLs. See Section 3.2 for details.

Figure 10. Same as in Fig.8, but with the PL zero points calibrated using DR2 parallaxes for the four Galactic RRLs and TGAS parallax for RR Lyr itself. See Section 3.2 for details.

Cep) to derive PL relations in the WISE MIR passbands directly calibrated on the HST parallaxes of the stars.

Similarly, Dambis et al. (2014) derived metallicity-dependent MIR PLZ relations, using WISE W1 observations for 360 RRLs in 15 MW GCs and W2 observations for 275 RRLs in nine GCs. The authors calibrated the zero points in two different ways: based on a statistical-parallax analysis and using the HST parallaxes of the four RRLs in Madore et al. (2013). The corresponding zero

points differ by more than 0.3 mag, with the statistical-parallax calibration providing fainter absolute magnitudes. PL relations in the WISE W1, W2, and W3 passbands were derived also by Klein et al. (2014) for RRab and RRc stars separately, using 129 field MW RRLs whose distances were inferred from the MV− [Fe/H]

relation in Chaboyer (1999) calibrated with Benedict et al. (2011)’s trigonometric parallaxes.

Neeley et al. (2015) derived PL relations in the 3.6 and 4.5 μm passbands, using 37 RRLs observed with Spitzer IRAC in the GC M4. The zero-point calibration was obtained combining the HST parallaxes with the Spitzer photometry of the five RRLs in Benedict et al. (2011) observed as part of the CRRP and, as an alternative, by adopting the distance to M4 from Braga et al. (2015). New theoretical PLZ relations in the Spitzer and WISE passbands were published by Neeley et al. (2017), who also revised the empirical relations in Neeley et al. (2015) using M4 data reduced with a new Spitzer pipeline.

Gaia Collaboration (2017) calibrated their own PL relation in the W1 passband using the TGAS parallaxes of 198 MW RRLs and adopting the slope from Madore et al. (2013). Gaia Collaboration (2017) used three different approaches to infer the zero point of the PL relations. In Table3we list the zero point derived using the ABL (Arenou & Luri1999), which is also the procedure adopted to calibrate the PL relations derived in this paper.

Sesar et al. (2017) studied a sample of about 100 MW RRab stars and calibrated the MIR W1 and W2 PLZ relations based on TGAS parallaxes. Recently, Muraveva et al. (2018) derived new W1-band PLZ relations using Gaia DR2 parallaxes for a sample of 401 MW RRL variables and for a reduced sample of 23 nearby RRLs whose metallicity was estimated from high-resolution spectroscopy.

The MIR relation obtained by Muraveva et al. (2018) using the sample of 23 nearby RRLs after correcting for a Gaia DR2 parallax offset of 0.056 mas is shown in Table3. For ease of comparison in Table3we grouped the PL and PLZ relations in the W1 and 3.6 μm passbands, and those in the W2 and 4.5 μm bands, since they are very similar.

The slope of the PL relation in the 3.6 μm band derived in this paper is shallower than values published in previous studies but still consistent with them within the errors. The slope of the PL relation in 4.5 μm passband is in very good agreement, within the quoted uncertainties, with all previous studies. The absolute magnitudes based on the TGAS and DR2 parallaxes are in very good agreement to each other and generally fainter than those based on the HST parallaxes. In Section 4 we discuss the distance to Reticulum inferred from the new PL relations and their different calibration procedures.

4 D I S TA N C E T O R E T I C U L U M

Several independent estimates of the distance to Reticulum are available in the literature. Table4summarizes the distances de-rived in this study and other key works. We indicate the pho-tometric passbands and the reddening values used in the various analyses.

Demers & Kunkel (1976) derived a distance modulus for Reticu-lum of μ= 18.51 mag using the mean apparent V magnitudes of the HB stars. These authors also found that the mean B magnitude of the RRLs in Reticulum is about 0.15 mag brighter than that for RRLs in the LMC field. Several independent studies, such as Freedman et al. (2001) based on classical Cepheids, Clementini et al. (2003) using RRLs, and Pietrzy´nski et al. (2013) from a sample of eight eclipsing binaries, have shown that the distance to the LMC is μ∼ 18.50 mag

(see e.g de Grijs et al.2014for a compilation of literature values). Therefore, the 0.15 mag difference in mean B magnitude found by Demers & Kunkel (1976) implies μ∼ 18.35 mag for the distance modulus of Reticulum, assuming that the reddening and B absolute magnitude are the same in the two systems.

Walker (1992), Ripepi et al. (2004), Mackey & Gilmore (2004), and Kuehn et al. (2013) derived the distance to Reticulum from the V-band observations of the 32 RRLs in the cluster. Kuehn et al. (2013) also derived a distance modulus for Reticulum using their I-band photometry of the cluster RRL variables, individual metal-licities estimated from the Fourier parameters of the light curves and adopting the PLZ relation of Catelan et al. (2004). Jeon et al. (2014) derived the distance to Reticulum by fitting theoretical isochrones to the cluster V versus (B− V) CMD. All distance moduli based on the V photometry agree on a distance modulus value around μ= 18.40 mag, except Demers & Kunkel (1976)’s that is about 0.1 mag larger.

Dall’Ora et al. (2004) combined the apparent K-band magnitudes of 30 RRLs in Reticulum with the theoretical PLZ relation by Bono et al. (2003) obtaining the distance modulus μ= 18.52 ± 0.005 (random)± 0.117 (systematic) mag, which is about 0.1 mag larger than that derived from the V photometry. These authors made a detailed analysis of possible systematic uncertainties affecting their analysis. Distances obtained from the NIR PLZ relation were found to be systematically larger than distances derived from Baade– Wesselink studies (Fernley1994), with the difference ranging from δμ= 0.10 mag for more metal-rich clusters to δμ = 0.32 mag for more metal-poor clusters. Therefore, the distance modulus of μ= 18.52 mag derived from the K-band photometry of Reticulum could be an overestimate. Later, Sollima et al. (2008) also found that the distance to Reticulum derived from the K-band photometry of Dall’Ora et al. (2004) is 0.04 mag larger than the distance inferred from the V-band photometry of Walker (1992).

A comparison of the values in Table4indeed confirms that dis-tances based on K- and I-band photometry are in general larger than distances derived from the optical photometry, which could sug-gest some issues in the zero-point calibrations in the two different passbands.

In this study we obtained distance moduli for Reticulum by apply-ing our PL relations to estimate individual distances to each of the 24 and 23 RRLs used in our analysis of the 3.6 μm and 4.5 μm PL re-lations, respectively, then averaging the resulting values. Errors are the standard deviations from the average. We obtained the follow-ing distance moduli for Reticulum: μ[3.6]= 18.43 ± 0.06 mag and

μ[4.5]= 18.43 ± 0.08 mag from the PL relations calibrated on the

HST parallaxes; μ[3.6]= 18.33 ± 0.06 mag and μ[4.5]= 18.34 ± 0.08

mag using the TGAS parallaxes and μ[3.6]= 18.32 ± 0.06 mag and

μ[4.5]= 18.34 ± 0.08 mag using the DR2 parallaxes. These values

are also reported in the bottom part of Table4. The distance moduli estimated using the PL relations calibrated on the HST parallaxes are in good agreement with the majority of the literature estimates, especially with those based on the V photometry. Distance moduli obtained from the PL relations calibrated on the TGAS and DR2 parallaxes are generally smaller, but still consistent within errors with the majority of previous estimates.

Pietrzy´nski et al. (2013) estimated a distance to the LMC of D= 49.97 ± 0.19 (stat) ± 1.11 (syst) kpc (μ0= 18.493 ± 0.008

(stat)± 0.047 (syst) mag) from eight long-period eclipsing binary systems which are all located relatively close to the barycentre of the LMC. Taking the mean value of all six distance moduli derived in this study, we conclude that Reticulum is located approximately 3 kpc closer to us than the barycentre of the LMC.

Table 4. Comparison of the distance moduli for Reticulum derived in this study and literature values.

Reference Band E(B− V) Distance modulus

(mag) (mag)

Demers & Kunkel (1976) V 0.02 18.51

Walker (1992) V 0.03 18.38

Ripepi et al. (2004) V 0.02 18.39± 0.12

Mackey & Gilmore (2004) V 0.05 18.39± 0.12

Kuehn et al. (2013) V 0.016 18.40± 0.20 Jeon et al. (2014) V 0.03 18.40 Sollima et al. (2008) V 0.03 18.44± 0.14 Kuehn et al. (2013) I 0.016 18.47± 0.06 Dall’Ora et al. (2004) K 0.03 18.52± 0.05 ± 0.117 Sollima et al. (2008) K 0.03 18.48± 0.11 This paper (HST) [3.6] 0.03 18.43± 0.06

This paper (TGAS) [3.6] 0.03 18.33± 0.06

This paper (DR2) [3.6] 0.03 18.32± 0.06

This paper (HST) [4.5] 0.03 18.43± 0.08

This paper (TGAS) [4.5] 0.03 18.34± 0.08

This paper (DR2) [4.5] 0.03 18.34± 0.08

5 S U M M A RY

We have analysed a sample of 30 RRLs in the LMC cluster Retic-ulum that we observed in the 3.6 and 4.5 μm passbands with IRAC in the Spitzer Warm Cycle 9 as part of the CRRP. Observations consist of 12 epochs over an∼14 h time span, allowing a good cov-erage of the RRL light curves. We analysed the light curves with the GRATIS software to derive MIR characteristic parameters (ampli-tudes and intensity-averaged mean magni(ampli-tudes). We combined our MIR data with optical photometry from Jeon et al. (2014), and the resulting CMDs show a well-defined RGB, an HB well populated by the cluster RRLs, and a few variables with magnitude/colour affected by crowding. Based on our analysis of the light curves, the position on the CMD and the appearance on the images we selected a sample of 24 RRLs that we used to fit the MIR PL relations. The slope of the PL relation in 3.6 μm passband derived in this paper (−2.15 ± 0.23) is shallower than the values published in previous studies but still consistent within the errors. The slope of the PL relation in the 4.5 μm passband (−2.47 ± 0.32), based on the 23 RRLs remained after the 3σ clipping procedure, is in very good agreement, within the errors, with the literature studies.

To estimate the zero points of our MIR PL relations, we used five MW RRLs for which trigonometric parallaxes were measured with the HST FGS (Benedict et al.2011), Gaia as part of TGAS in Gaia DR1 and Gaia DR2, based only on Gaia measurements, allowing us to make a direct comparison of the three calibrations. We used the ABL method to fit the MIR RRL PL relations, as this operates directly in parallax space, avoiding non-symmetrical errors in absolute magnitudes caused by inversion of the parallaxes. We found that the absolute magnitudes based on the TGAS and DR2 parallaxes are in good agreement with each other and are in general fainter than those based on the HST parallaxes. However, the sample of RRLs used in the analysis is too small to draw any conclusion about possible systematic offsets.

We obtained the following distance moduli for Reticulum: μ[3.6]= 18.43 ± 0.06 mag and μ[4.5]= 18.43 ± 0.08 mag from the

PL relations calibrated on the HST parallaxes; μ[3.6]= 18.33 ± 0.06

mag and μ[4.5]= 18.34 ± 0.08 mag using the TGAS parallaxes and

μ[3.6]= 18.32 ± 0.06 mag and μ[4.5]= 18.34 ± 0.08 mag using the

DR2 parallaxes. These distance moduli are in good agreement with

the literature values estimated using the V band photometry, while the agreement with the distance values of Reticulum based on the I and K photometry is less pronounced.

Based on the mean of the distance moduli derived in this study, we conclude that Reticulum is located approximately 3 kpc closer to us than the barycentre of the LMC.

AC K N OW L E D G E M E N T S

We thank the anonymous reviewer for a careful reading and valu-able suggestions that have improved the paper. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech and makes use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https: //www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. Sup-port to this study has been provided by PRIN-INAF2014, “EXCAL-IBUR’S” (P.I. G. Clementini), from the Agenzia Spaziale Italiana (ASI) through grants ASI I/058/10/0 and ASI 2014- 025-R.1.2015 and by Premiale 2015, “MITiC” (P.I. B. Garilli). GC thanks The Carnegie Observatories visitor programme for support as science visitor.

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S U P P O RT I N G I N F O R M AT I O N

Supplementary data are available atMNRASonline.

Table1 full new.txt

Table 1. Main properties of the 364 sources in common between

our MIR catalogues and the visual catalogues of Jeon et al. (2014). Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.